A179190
Coefficient [x^n] of the Maclaurin series for 2 - sqrt(1 - 4*x - 4*x^2).
Original entry on oeis.org
1, 2, 4, 8, 24, 80, 288, 1088, 4256, 17088, 70016, 291584, 1230592, 5251584, 22623232, 98248704, 429677056, 1890700288, 8364824576, 37186449408, 166030266368, 744180244480, 3347321831424, 15104525959168, 68357598756864
Offset: 0
The Maclaurin series is 1 + 2*x + 4*x^2 + 8*x^3 + 24*x^4 + ...
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A179190 := proc(n) if n = 0 then 1; else add( doublefactorial(2*n-2*k-3) *2^(n-k) / k! / (n-2*k)!, k=0..floor(n/2)) ; end if; end proc: # R. J. Mathar, Jul 11 2011
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Table[SeriesCoefficient[Series[2-Sqrt[1-4*t-4*t^2], {t,0,n}], n], {n, 0, 30}] (* G. C. Greubel, Jan 25 2019 *)
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makelist(coeff(taylor(2-sqrt(1-4*x-4*x^2), x, 0, n), x, n), n, 0, 24); /* Bruno Berselli, Jul 04 2011 */
A260774
Certain directed lattice paths.
Original entry on oeis.org
1, 6, 33, 189, 1107, 6588, 39663, 240894, 1473147, 9058554, 55954395, 346934745, 2157989445, 13459891500, 84152389833, 527224251861, 3309194474451, 20804569738218, 130987600581699, 825796890644895, 5212349717906889, 32935490120006604, 208316726580941037
Offset: 0
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b:= proc(x, y) option remember; `if`([x, y]=[0$2], 1,
`if`(x>0, add(b(x-1, y+j), j=-1..1), 0)+
`if`(y>0, b(x, y-1), 0)+`if`(y<0, b(x, y+1), 0))
end:
a:= n-> b(n, 1):
seq(a(n), n=0..23); # Alois P. Heinz, Sep 21 2021
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b[x_, y_] := b[x, y] = If[{x, y} == {0, 0}, 1,
If[x > 0, Sum[b[x - 1, y + j], {j, -1, 1}], 0] +
If[y > 0, b[x, y - 1], 0] + If[y < 0, b[x, y + 1], 0]];
a[n_] := b[n, 1];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, May 02 2022, after Alois P. Heinz *)
A260772
Certain directed lattice paths.
Original entry on oeis.org
1, 3, 10, 41, 190, 946, 4940, 26693, 147990, 837102, 4811860, 28027210, 165057100, 981177060, 5879570200, 35478788269, 215398416870, 1314794380374, 8064119033220, 49673222082782, 307163049317540, 1906066361809148, 11865666767361960, 74081851132379426
Offset: 0
- Lars Blomberg, Table of n, a(n) for n = 0..100
- M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
- M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, Discrete Mathematics, Volume 339, Issue 3, 6 March 2016, Pages 1116-1139.
- Heba Bou KaedBey, Mark van Hoeij, and Man Cheung Tsui, Solving Third Order Linear Difference Equations in Terms of Second Order Equations, arXiv:2402.11121 [math.AC], 2024. See p. 3.
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# A260772 satisfies a 4th-order recurrence that can be reduced
# to a 2nd-order recurrence given in this program t:
t := proc(n) options remember;
if n <= 1 then
[-1/2, 0, 1, 4][2*n+2]
else
(16*(n-2)*(2*n-3)*(5*n-2)*t(n-2) + (440*n^3-1056*n^2+724*n-144)*t(n-1))
/( n*(2*n+1)*(5*n-7) )
fi
end:
A260772 := proc(n)
t(n/2) + ( (2-2*n)*t((n-1)/2)+(n+2)*t((n+1)/2) ) / (1+5*n)
end:
seq(A260772(i),i=0..100);
# Mark van Hoeij, Jul 14 2022
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a(n):=if n=0 then 1 else sum((-1)^j*binomial(n,j)*binomial(3*n-4*j,n-4*j+1),j,0,(n+1)/4)/n; /* Vladimir Kruchinin, Apr 04 2019 */
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a(n) = if (n==0, 1, sum(j=0, (n+1)/4, (-1)^j*binomial(n,j)*binomial(3*n-4*j, n-4*j+1))/n); \\ Michel Marcus, Apr 05 2019
Showing 1-3 of 3 results.
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